On 3-submanifolds of S3 which admit complete spanning curve systems

Let M be a compact connected 3-submanifold of the 3-sphere S3 with one boundary component F such that there exists a collection of n pairwise disjoint connected orientable surfaces S = {S1, · · ·, Sn} properly embedded in M, ∂S = {∂S1, · · ·, ∂Sn} is a complete curve system on F. We call S a complete surface system for M, and ∂S a complete spanning curve system for M. In the present paper, the authors show that the equivalent classes of complete spanning curve systems for M are unique, that is, any complete spanning curve system for M is equivalent to ∂S. As an application of the result, it is shown that the image of the natural homomorphism from the mapping class group M(M) to M(F) is a subgroup of the handlebody subgroup Hn.